## Involutions on the affine Grassmannian and moduli spaces of principal bundles

### Anthony Henderson

#### Abstract

Let $$G$$ be a connected reductive group over $$\mathbb{C}$$. We show that a certain involution of an open subset of the affine Grassmannian of $$G$$, defined previously by Achar and the author, corresponds to the action of the nontrivial Weyl group element of $$\mathrm{SL}(2)$$ on the framed moduli space of $$\mathbb{G}_m$$-equivariant principal $$G$$-bundles on $$\mathbb{P}^2$$. As a result, the fixed-point set of the involution can be partitioned into strata indexed by conjugacy classes of homomorphisms $$N\to G$$ where $$N$$ is the normalizer of $$\mathbb{G}_m$$ in $$\mathrm{SL}(2)$$. In the case where $$G=\mathrm{GL}(r)$$, the strata are Nakajima quiver varieties $$\mathfrak{M}_0^{\mathrm{reg}}(\mathbf{v},\mathbf{w})$$ of type $$D$$.

Keywords: Affine Grassmannian; moduli space; quiver variety.

: Primary 14J60; Secondary 14M15, 17B08.

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 Thursday, December 17, 2015